Bilevel optimization has recently attracted growing interests due to its wide applications in modern machine learning problems. Although recent studies have characterized the convergence rate for several such popular algorithms, it is still unclear how much further these convergence rates can be improved. In this paper, we address this fundamental question from two perspectives. First, we provide the first-known lower complexity bounds of $\widetilde{\Omega}(\frac{1}{\sqrt{\mu_x}\mu_y})$ and $\widetilde \Omega\big(\frac{1}{\sqrt{\epsilon}}\min\{\frac{1}{\mu_y},\frac{1}{\sqrt{\epsilon^{3}}}\}\big)$ respectively for strongly-convex-strongly-convex and convex-strongly-convex bilevel optimizations. Second, we propose an accelerated bilevel optimizer named AccBiO, whose complexity improves the existing upper bounds orderwisely under strongly-convex-strongly-convex, convex-strongly-convex and nonconvex-strongly-convex geometries. We further show that AccBiO achieves the optimal results (i.e., the upper and lower bounds match) under certain conditions up to logarithmic factors. Interestingly, our lower bounds under both geometries are larger than the corresponding optimal complexities of minimax optimization, establishing that bilevel optimization is provably more challenging than minimax optimization. We finally discuss the extensions and applications of our results to other problems such as minimax optimization.

Although model-agnostic meta-learning (MAML) is a very successful algorithm in meta-learning practice, it can have high computational cost because it updates all model parameters over both the inner loop of task-specific adaptation and the outer-loop of meta initialization training. A more efficient algorithm ANIL (which refers to almost no inner loop) was proposed recently by Raghu et al. 2019, which adapts only a small subset of parameters in the inner loop and thus has substantially less computational cost than MAML as demonstrated by extensive experiments. However, the theoretical convergence of ANIL has not been studied yet. In this paper, we characterize the convergence rate and the computational complexity for ANIL under two representative inner-loop loss geometries, i.e., strongly-convexity and nonconvexity. Our results show that such a geometric property can significantly affect the overall convergence performance of ANIL. For example, ANIL achieves a faster convergence rate for a strongly-convex inner-loop loss as the number $N$ of inner-loop gradient descent steps increases, but a slower convergence rate for a nonconvex inner-loop loss as $N$ increases. Moreover, our complexity analysis provides a theoretical quantification on the improved efficiency of ANIL over MAML. The experiments on standard few-shot meta-learning benchmarks validate our theoretical findings.

Bilevel optimization has arisen as a powerful tool for many machine learning problems such as meta-learning, hyper-parameter optimization, reinforcement learning, etc. In this paper, we investigate the nonconvex-strongly-convex bilevel optimization problem, and propose two novel algorithms named deterBiO and stocBiO respectively for the deterministic and stochastic settings. At the core design of deterBiO is the construction of a low-cost and easy-to-implement hyper-gradient estimator via a simple back-propagation. In addition, stocBiO updates with the mini-batch data sampling rather than the existing single-sample schemes, where a sample-efficient Hessian inverse estimator is proposed. We provide the finite-time convergence guarantee for both algorithms, and show that they outperform the best known computational complexities orderwisely with respect to the condition number $\kappa$ and/or the target accuracy $\epsilon$. We further demonstrate the superior efficiency of the proposed algorithms by the experiments on meta-learning and hyper-parameter optimization.

Many existing zeroth-order optimization (ZO) algorithms adopt two-point feedback schemes due to their fast convergence rate compared to one-point feedback schemes. However, two-point schemes require two evaluations of the objective function at each iteration, which can be impractical in applications where the data are not all available a priori, e.g., in online optimization. In this paper, we propose a novel one-point feedback scheme that queries the function value only once at each iteration and estimates the gradient using the residual between two consecutive feedback points. When optimizing a deterministic Lipschitz function, we show that the query complexity of ZO with the proposed one-point residual feedback matches that of ZO with the existing two-point feedback schemes. Moreover, the query complexity of the proposed algorithm can be improved when the objective function has Lipschitz gradient. Then, for stochastic bandit optimization problems, we show that ZO with one-point residual feedback achieves the same convergence rate as that of ZO with two-point feedback with uncontrollable data samples. We demonstrate the effectiveness of the proposed one-point residual feedback via extensive numerical experiments.

As a popular meta-learning approach, the model-agnostic meta-learning (MAML) algorithm has been widely used due to its simplicity and effectiveness. However, the convergence of the general multi-step MAML still remains unexplored. In this paper, we develop a new theoretical framework, under which we characterize the convergence rate and the computational complexity of multi-step MAML. Our results indicate that $N$-step MAML attains the convergence with linearly increasing complexity with $N$ under a properly chosen inner stepsize. We then take a further step to develop a more efficient Hessian-free MAML. We first show that the existing zeroth-order Hessian estimator contains a constant-level estimation error so that the MAML algorithm can perform unstably. To address this issue, we propose a novel Hessian estimator via a gradient-based Gaussian smoothing method, and show that it achieves a much smaller estimation bias and variance, and the resulting algorithm achieves the same performance guarantee as the original MAML under mild conditions. Our experiments validate our theory and demonstrate the effectiveness of the proposed Hessian estimator.

Two types of zeroth-order stochastic algorithms have recently been designed for nonconvex optimization respectively based on the first-order techniques SVRG and SARAH/SPIDER. This paper addresses several important issues that are still open in these methods. First, all existing SVRG-type zeroth-order algorithms suffer from worse function query complexities than either zeroth-order gradient descent (ZO-GD) or stochastic gradient descent (ZO-SGD). In this paper, we propose a new algorithm ZO-SVRG-Coord-Rand and develop a new analysis for an existing ZO-SVRG-Coord algorithm proposed in Liu et al. 2018b, and show that both ZO-SVRG-Coord-Rand and ZO-SVRG-Coord (under our new analysis) outperform other exiting SVRG-type zeroth-order methods as well as ZO-GD and ZO-SGD. Second, the existing SPIDER-type algorithm SPIDER-SZO (Fang et al. 2018) has superior theoretical performance, but suffers from the generation of a large number of Gaussian random variables as well as a $\sqrt{\epsilon}$-level stepsize in practice. In this paper, we develop a new algorithm ZO-SPIDER-Coord, which is free from Gaussian variable generation and allows a large constant stepsize while maintaining the same convergence rate and query complexity, and we further show that ZO-SPIDER-Coord automatically achieves a linear convergence rate as the iterate enters into a local PL region without restart and algorithmic modification.

An important class of distance metrics proposed for training generative adversarial networks (GANs) is the integral probability metric (IPM), in which the neural net distance captures the practical GAN training via two neural networks. This paper investigates the minimax estimation problem of the neural net distance based on samples drawn from the distributions. We develop the first known minimax lower bound on the estimation error of the neural net distance, and an upper bound tighter than an existing bound on the estimator error for the empirical neural net distance. Our lower and upper bounds match not only in the order of the sample size but also in terms of the norm of the parameter matrices of neural networks, which justifies the empirical neural net distance as a good approximation of the true neural net distance for training GANs in practice.

An important class of distance metrics proposed for training generative adversarial networks (GANs) is the integral probability metric (IPM), in which the neural net distance captures the practical GAN training via two neural networks. This paper investigates the minimax estimation problem of the neural net distance based on samples drawn from the distributions. We develop the first known minimax lower bound on the estimation error of the neural net distance, and an upper bound tighter than an existing bound on the estimator error for the empirical neural net distance. Our lower and upper bounds match not only in the order of the sample size but also in terms of the norm of the parameter matrices of neural networks, which justifies the empirical neural net distance as a good approximation of the true neural net distance for training GANs in practice.

An important class of distance metrics proposed for training generative adversarial networks (GANs) is the integral probability metric (IPM), in which the neural net distance captures the practical GAN training via two neural networks. This paper investigates the minimax estimation problem of the neural net distance based on samples drawn from the distributions. We develop the first known minimax lower bound on the estimation error of the neural net distance, and an upper bound tighter than an existing bound on the estimator error for the empirical neural net distance. Our lower and upper bounds match not only in the order of the sample size but also in terms of the norm of the parameter matrices of neural networks, which justifies the empirical neural net distance as a good approximation of the true neural net distance for training GANs in practice.

There has been extensive research on developing stochastic variance reduced methods to solve large-scale optimization problems. More recently, a novel algorithm of such a type named SPIDER has been developed in \cite{Fang2018}, which was shown to outperform existing algorithms of the same type and meet the lower bound in certain regimes. Though interesting in theory, SPIDER requires $\epsilon$-level stepsize to guarantee the convergence, and consequently runs slow in practice. This paper proposes SpiderBoost as an improved SPIDER scheme, which comes with two major advantages compared to SPIDER. First, it allows much larger stepsize without sacrificing the convergence rate, and hence runs substantially faster than SPIDER in practice. Second, it extends much more easily to proximal algorithms with guaranteed convergence for solving composite optimization problems, which appears challenging for SPIDER due to stringent requirement on per-iteration increment to guarantee its convergence. Both advantages can be attributed to the new convergence analysis we develop for SpiderBoost that allows much more flexibility for choosing algorithm parameters. As further generalization of SpiderBoost, we show that proximal SpiderBoost achieves a stochastic first-order oracle (SFO) complexity of $\mathcal{O}(\min\{n^{1/2}\epsilon^{-1},\epsilon^{-3/2}\})$ for composite optimization, which improves the existing best results by a factor of $\mathcal{O}(\min\{n^{1/6},\epsilon^{-1/6}\})$.